NCERT Class 11-Math's: Chapter –14 Mathematical Reasoning Part 3

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Example 12: Write the negation of each of the following disjunction:

(a) Ram is in Class X or Rahim is in Class XII.

(b) is greater than or is less than .

Answer:

(a) Let : Ram is in Class X and : Rahim is in Class XII.

Then the disjunction in is given by .

Now : Ram is not in Class X.

: Rahim is not in Class XII.

Then, using , negation of is given by

: Ram is not in Class X and Rahim is not in Class XII.

(b) Write is greater than 4, and is less than .

Then, using , negation of is given by

is not greater than and is not less than .

14.1.11 Negation of a negation As already remarked the negation is not a connective but a modifier. It only modifies a given statement and applies only to a single simple statement. Therefore, in view of and , for a statement p, we have

Negation of negation of a statement is the statement itself. Equivalently, we write

14.1.12 The conditional statement Recall that if p and q are any two statements, then the compound statement “if p then q” formed by joining p and q by a connective ‘if then’ is called a conditional statement or an implication and is written in symbolic form as or . Here, p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement

Remark The conditional statement can be expressed in several different ways. Some of the common expressions are:

(a) If , then

(b) if

(c) only if

(d) is sufficient for

(e) is necessary for .

Observe that the conditional statement reflects the idea that whenever it is known that is true, it will have to follow that is also true.

Example 13: Each of the following statements is also a conditional statement.

(i) If , then Rekha will get an ice-cream.

(ii) If you eat your dinner, then you will get dessert.

(iii) If John works hard, then it will rain today.

(iv) If ABC is a triangle, then .

Example 14: Express in English, the statement , where

p: it is raining today

Solution: The required conditional statement is “If it is raining today, then

14.1.13 Contrapositive of a conditional statement: The statement “” is called the contrapositive of the statement

Example 15: Write each of the following statements in its equivalent contrapositive form:

(i) If my car is in the repair shop, then I cannot go to the market.

(ii) If Karim cannot swim to the fort, then he cannot swim across the river.

Solution: (i) Let “p : my car is in the repair shop” and “ can-not go to the market”.

Then, the given statement in symbolic form is pq. Therefore, its contrapositive is given by .

Now : My car is not in the repair shop.

and : I can go to the market

Therefore, the contrapositive of the given statement is

“If I can go to the market, then my car is not in the repair shop”.

(ii) Proceeding on the lines of the solution of (i), the contrapositive of the statement in (ii) is

“If Karim can swim across the river, then he can swim to the fort”.

14.1.14 Converse of a conditional statement: The conditional statement “” is called the converse of the conditional statement “

Example 16: Write the converse of the following statements

(i) If , then

(ii) If ABC is an equilateral triangle, then ABC is an isosceles triangle

Solution: (i) Let

Therefore, the converse of the statement is given by

“If , then

(ii) Converse of the given statement is

“If ABC is an isosceles triangle, then ABC is an equilateral triangle.”